Answer :

To solve the equation [tex]\(-2x^4 + 9x^2 + 18x = -3x^4 - 2x^3\)[/tex], let’s follow these steps:

1. Move all terms to one side of the equation:

Begin by adding [tex]\(3x^4 + 2x^3\)[/tex] to both sides to collect all terms on one side of the equation.

[tex]\[
-2x^4 + 9x^2 + 18x + 3x^4 + 2x^3 = 0
\][/tex]

2. Combine like terms:

Simplify the equation by combining like terms.

[tex]\[
(-2x^4 + 3x^4) + 2x^3 + 9x^2 + 18x = 0
\][/tex]

This simplifies to:

[tex]\[
x^4 + 2x^3 + 9x^2 + 18x = 0
\][/tex]

3. Factor out common factors:

Notice that [tex]\(x\)[/tex] is a common factor in all terms.

[tex]\[
x(x^3 + 2x^2 + 9x + 18) = 0
\][/tex]

4. Solve for the solutions:

Since [tex]\(x\)[/tex] is factored out, one solution is:

[tex]\[
x = 0
\][/tex]

Now solve the cubic equation:

[tex]\[
x^3 + 2x^2 + 9x + 18 = 0
\][/tex]

5. Finding roots of the cubic polynomial:

The cubic equation can have either one real root and two non-real complex roots or all three being real. Through further analysis or solving techniques, we find that:

[tex]\[
x = -2, \quad x = -3i, \quad x = 3i
\][/tex]

Thus, the solutions to the original equation are:

[tex]\[
x = 0, \quad x = -2, \quad x = -3i, \quad x = 3i
\][/tex]

These include one real solution ([tex]\(-2\)[/tex]), one repeated real solution ([tex]\(0\)[/tex]), and two complex solutions ([tex]\(-3i\)[/tex] and [tex]\(3i\)[/tex]).

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Rewritten by : Barada